B-decay Anomalies in a Composite Leptoquark Model

The collection of a few anomalies in semileptonic $B$-decays, especially in $b\rightarrow c \tau \bar{\nu}$, invites to speculate about the emergence of some striking new phenomena, perhaps interpretable in terms of a weakly broken $U(2)^n$ flavor symmetry and of leptoquark mediators. Here we aim at a partial UV completion of this interpretation by generalizing the minimal composite Higgs model to include a composite vector leptoquark as well.


I. INTRODUCTION
A number of anomalies in the decays of B mesons continue to receive much attention.
As recalled below, the statistically most significant among these anomalies is of special interest since, at the partonic level, b → cτν, it involves three third generation particles.
As such it is suggestive of an explanation in terms of a U (2) n flavor symmetry that distinguishes between the third generation of fermions as singlets and the first two generations as doublets [1].
Within this context ref. [2] looked at the ability of leptoquark models, in particular spinone leptoquarks, to explain some of these anomalies. However a model with massive vector fields cries out for a UV completion (see e.g. [3]). This is particularly true since, as one can anticipate from the relatively large size of the putative deviation from the Standard Model (SM) tree level amplitude, a fairly large coupling of the leptoquark must be invoked. The aim of this paper is to investigate whether it is possible to make a composite model that can serve as a (more) UV complete explanation of these flavor anomalies. In particular, we are looking to generalize the simplified Composite Higgs Models (CHM) of [4] to a case which includes leptoquarks. To this end we extend the global symmetry group of the strong sector from SU (3) × SO(5) × U (1) [5] to SU (4) × SO(5) × U (1), where SU (4) is the Pati-Salam group. The extension from SU (3) to SU (4) can be seen as natural if one thinks of composite leptons as necessary to give masses to the standard leptons by bilinear mixing, as in the quark case often discussed.
The experimental measurements of interest include a combined 4.0σ excess over the SM, which is seen by three experiments in the charged current process with = e, µ. Assuming a common scaling of R D and R D * with respect to their SM predictions, a one parameter fit to the averages presented by the Heavy Flavor Averaging Group (HFAG) yields R D ( * ) /(R D ( * ) ) SM = 1.27 ± 0.06 [6,7]. The HFAG result makes use of experimental measurements from BaBar [8,9], LHCb [10], and Belle [11,12] (see also [13]); as well as the theoretical predictions of refs. [14,15] (see also [16,17]).
Furthermore, LHCb has reported [18,19] a 2.6σ deviation from the SM in the neutral current process possibly indicating a violation of lepton flavor universality (LFU). Specifically, for M 2 + − ∈ [1, 6] GeV 2 the measured value of R K is 0.745 +0.090 −0.074 (stat) ± 0.035(syst), compared to a SM value that is close to 1 [20,21]. Global fits to all b → s data seem to indicate a more general tension with the SM [22,23]. 1 However, many of these observables are subject to significant hadronic uncertainties, whereas R K and R D ( * ) , are not.
Ref. [48] is especially of interest in light of the goal of this work as it challenges the idea that the B decay anomalies could be due to simple extensions of the SM, i.e. a single leptoquark field. Specifically, when only the minimal set of operators needed to explain R D ( * ) and R K are generated at some scale Λ v, the RG evolution of these operators generates unacceptably large deviations from lepton flavor universality in Z and τ decays as well as lepton flavor violating τ decays. The particular operators are Q (1) q and Q q ; see [49,50] for notation and the explicit form of the RGE. While a full one-loop RGE analysis is beyond the scope of this work, we note there are at least two effects that distinguish the model under consideration in this work from that of ref. [48]. The first is that there are more dimension-six operators than the two listed above, which are generated at tree level that contribute to the relevant RGE, e.g. Q and Q H , as well operators that do not contribute to the RGE of interest. Some of the additional operators contribute to the RGE with the opposite sign of the contribution coming from the operators considered in [48]. Secondly there are direct contributions to the observables of interest that are generated at the scale Λ at the one-loop level. Though these contributions do not have a 1 A recent update of [22] claims the combined tension with the SM has increased to 4.5σ [24]. log-enhancement as the RGE contributions do, they can still serve to partially cancel the effects of the RGE contributions.
The rest of this paper is organized as follows. Sec. II describes the field content of the model as well as the mass spectra and mixing angles associated with the fermions and vector bosons. The tree level amplitudes and viable parameter space are presented in sec. III. This is followed by a discussion of electroweak precision data in sec. IV. Then in sec. V a description is given of a number of features of this model, which distinguish it from the usual partial CHM. Finally, our conclusions are given in sec. VI.

II. PARTICLE CONTENT
We start by describing the field content of the composite sector in terms of its representations under the unbroken global symmetry of the composite sector, where T A=1,...,15 are the generators of SU (4) with normalization Tr(T A T B ) = δ AB /2. The coefficient in front of T 15 in eq. (3) is necessary to get the correct hypercharge, since 2/3 T 15 = (B − L)/2, with B and L being baryon and lepton numbers respectively.

A. Vector Boson Masses and Mixings
The vector boson masses and mixings are analogous to those of ref. [4] or to those of a two site model of the standard CHM, apart from two main differences: we have to include SU (4) instead of SU (3) and the elementary weak hypercharge gauge boson mixes with three composite fields (associated with T 15 , T 3 R and X). The SU (4) composite bosons can be written as where λ a=1,...  With the only purpose of simplifying the formulae in the following we take g ρR = g X and M ρR = M X . On the other hand, before mixing, the elementary fields associated with the with their own couplings {g e3 , g e2 , g e1 }.
After mixing (and prior to electroweak symmetry breaking), the mass eigenstates are superpositions of the states in (4), (5), (6). Of special interest to us are the leptoquarks, V µ , V † µ , which stay unmixed, and the totally neutral states, in number four, one of which, 2, stays also unmixed. 2 In the following we shall take and present results at leading order in Defining B * µ = (W R3 µ + V X µ )/ √ 2 and calling ρ cµ and ρ eµ the collections of composite and elementary vectors respectively, the Lagrangian for the gauge sector is The mass eigenstates (prior to electroweak symmetry breaking) are related to the states in eq. (8) by where the mixing angles θ 2 and θ 3 are For simplicity we show the rotation matrix in the neutral sector at leading order in g e1 g ρR 1 and g e1 gρ 1. The physical gauge couplings are g 3 = g e3 c 3 = g ρ s 3 , g 2 = g e2 c 2 = g ρL s 2 and The mass spectrum in the gauge sector is where c 2,3 = cos θ 2,3 , c 15 ≈ 1 + O( g e1 gρ ) 2 and c R ≈ 1 + O( g e1 g ρR ) 2 . Since also θ 2,3 are bound to be small, one can consider to a reasonable approximation only three different masses involved, or in fact only two after imposing g ρR = g ρL to respect custodial symmetry and Note that The part of the Lagrangian that describes the leptoquark interactions with the SM gauge fields is given by From the above interaction terms one finds that the leptoquark V µ couples to the SM fields G a µ and B µ as in [2] with k s = k Y = 1. This means that potentially dangerous contributions to dipole operators (responsible for µ → eγ, τ → µγ decays, etc.) are finite in our model, unlike in the general case of the low energy leptoquark model [2].
Our notation for the bidoublet model is where β = 1, 2, 3 is a fundamental color index. The components of ψ ± are further reduced 3 We adopt the nomenclature of ref. [51]. as where in the right-hand side of these equations we make explicit the transformation properties of the various components under the SM gauge group. All the X states are exotic with their charge explicitly indicated, while their SU (3) properties are left understood.
Following ref. [51] we attribute the basic distinction between the third and the lighter first and second generations to the presence of an approximate U (2) n flavor symmetry which is unbroken in the composite sector and is weakly broken along specific "spurion" directions only in the mass mixings between the elementary and the composite fermions. In particular, to avoid unobserved flavor-breaking effects, we rely on the idea of left-or right-compositeness [52][53][54]. In the present context left-compositeness and right-compositeness can be implemented invoking as intermediate symmetries or respectively. In particular, in the case of right-compositeness, one ends up with flavor violation in the up quark sector suppressed by inverse power of z 3 ≡ s Lu3 /s Ld3 , as defined below, which is required to be large by consistency with the Zb LbL coupling measurements (see sec. IV). This, in turn, suppresses the contribution to the charged-current B anomaly, making impossible to reproduce the observed deviation. Therefore, in the following we will consider only left-compositeness.
The Yukawa and mass terms for the fermionic resonances in the strong sector are given by L bidoublet where Y ± , m ψ ± and m χ ± are U (2) preserving flavor diagonal matrices, so that Y T ± = (Y ±3 , Y ±2 , Y ±2 ), and similarly for m Ψ± and m χ± . As in ref. [51] the quark mixing Lagrangian is given by Similarly the lepton mixing Lagrangian is The mixings in the first lines of (21) and (22) break the symmetry of the strong sector down to G LC . This symmetry is in turn broken minimally by the spurions in the second lines of the same equations.
The SM Yukawa couplings for up and down quarks can be written in terms of the spurions as in [51]. Adopting also the same definitions as in [51] for the mixings s L , s R between the elementary and the composite fermions, it iŝ while for the charged lepton we obtain where and similarly for y b and a d with the obvious replacements. Extending The parameter x b is not determined by CKM data and it enters in flavor violating terms in up and down quark sector via respectively.
Here we are not concerned with neutrino masses and mixings, which can arise from a suitable Majorana mass matrix of the right handed neutrinos mixed with the compositẽ N states. In any event, to an excellent level of approximation, we can study B anomalies in the basis of neutrino current-eigenstates, where the charged current leptonic weak interactions are flavor-diagonal.

III. TREE LEVEL AMPLITUDES FOR B ANOMALIES
Exchanges of spin-one resonances contribute to tree level b → cτ ν and b → s decays as well as to ∆F = 2 transitions. The interaction Lagrangian of the composite vectors with the elementary quarks and leptons in the mass basis is given in appendix B. We shall neglect terms suppressed by 1/z 3 = s Ld3 /s Lu3 and 1/z 3e = s Le3 /s Lν3 as z 3 , z 3e are required to be large to control the deviations from the SM of the Zb LbL and Zτ LτL couplings respectively. It is also convenient to define the following quantity Contributions to the operator (c L γ µ b L )(τ L γ µ ν 3L ) arise from the t-channel exchange of the leptoquark V µ and the s-channel exchanges of W H± . For b → cτν 3 one has where Lu3 A Lν3 .
For small θ 2 , f W * tends towards one, so that to explain the R D ( * ) anomaly at 1σ one needs For the neutral current process b → sµµ, there are leading contributions from the t-channel exchange of the leptoquark V µ and the s-channel exchanges of W H3 ,Ṽ and X.
One finds Tree level ∆F = 2 transitions are mediated by composite gluons G H , and composite electroweak vectors W H3 ,Ṽ , and X. In particular, for ∆B s = 2 one has where the f functions are the same as in (33) with θ Lν3 → θ Lu3 , and f G H = f W * with θ 2 → θ 3 . Neglecting the vector mixing angles in the f functions, as previously done, The plots in fig. 1 show the parameter space needed to explain R D ( * ) and R K as well as the parameter space consistent with measurements of ∆B s = 2 processes. 4 The range of In the oblique corrections at one-loop no new contributions arise to the S, T, U parameters from exchanges of the leptoquark, which is a singlet of SU (2) L (as is the case for all the neutral composite vectors). The only effect of the leptoquark is a contribution to the Y parameter [55], which is UV-sensitive and will have to be cutoff at a scale Λ by the composite dynamics. From the Lagrangian (15) one obtains [2] The strongest bound comes from δg Lτ , enhanced by a color factor of 3 with respect to δg Lb , which requires g ρ > 2 ÷ 3 for Λ close to maximal. • There is a vector singlet composite leptoquark, V µ , partly responsible for the anomalies in B-decays, with branching ratios likely close to 50% for bτ and tν τ . V µ can be directly searched in QCD pair production. Its exchange in the t-channel also contributes to bb → τ + τ − .
• There are exotic composite leptons with a mass within a few % degenerate with the exotic composite quarks that are normally discussed in the context of standard CHMs.
Before discussing in some details the first item, let us briefly comment on the second item. To the best of our knowledge so far there has been only one search for the pairproduction of spin-one leptoquarks decaying to third generation fermions at the LHC.
Potentially stronger bounds on vector leptoquark masses might be obtained by reinterpreting scalar leptoquark pair-production searches that used 8 or 13 TeV data. Table I summarizes Ref. [44] was the first to point out that bb → τ + τ − can be a signal of models that attempt to explain R D ( * ) . Subsequently ref. [45] throughly investigated bounds on expla- nations of R D ( * ) coming from new physics searches involving pairs of tau leptons. We will compare our results to those of [45] at the end of this subsection.
Let us first consider Z s alone. The total width of the Z generally includes decays to both pairs of third generation SM fermions as well as W + L W − L and Z L h. Decays to composite fermions, if allowed at all, are phase space suppressed, and we neglect them in what follows. Fig. 2 shows Γ Z /M Z as a function of M Z . All of the relevant formulas can be found in appendix C. The blue, orange, green, and red curves correspond toṼ µ , W H3 µ , X µ , and B H µ , respectively. The solid lines reproduce the central value of R D ( * ) assuming s Lu3 = s Lν3 and ξ = 0.1, and the shaded bands reproduce R D ( * ) at the 1σ level. We have not included a band for B H since its coupling to SM fermions in proportional to t 2 R , which leads to a feeble coupling.
The result of ATLAS at 8 TeV is the most constraining published experimental result on searches for new physics in τ + τ − [64]. CMS has released preliminary results at 13 TeV that are naïvely more constraining for M Z > ∼ 900 GeV [65]. However not enough information about cuts and efficiencies is provided to reinterpret this search in terms of the Z s of our model, which are not SM-like. We used the publicly available plot digitizer WebPlotDigitizer v3.10 [66] in performing this analysis. work we use the NNPDF collaboration's NNPDF23 lo as 0119 parton distribution function (PDF) grid [67,68]. To approximately match the cuts employed in the ATLAS search to those defined in eq. (C6) we take Y cut = 2.47, and p T cut = m tot T /2 where m tot T is defined in [64] for a given M Z . The blue, orange, and green curves correspond toṼ µ , W H3 µ , and X µ , respectively. We have not shown the B H µ limit, as the corresponding cross section is proportional to s 4 R , which essentially yields no bound. Just as in fig. 2 For X and W H3 these limits are robust, as their widths are less than and approximately equal to 20% of their mass at the implied limit, respectively. This is not the case forṼ , indicating that this limit is more uncertain.
Including only a single Z at a time may not be a good approximation to the full By looking at individual Z contributions, ref. [45] found that τ − τ + searches rule out some region of the space characterized by (M Z , Γ Z , |g b g τ |v 2 /M 2 Z ), where Γ Z is the total width and g b , g τ are the couplings of the Z to the b and the τ . For the Z s, a direct comparison is possible for ourṼ , W H and X bosons, using the total widths given in appendix C and the approximate relations We find good agreement between our results and the ones derivable from the figure 4 of [45]. 5 We did not directly investigate the bounds on the leptoquark of this work coming from its sole contribution to τ − τ + searches. However it is straightforward to translate the results of [45] into the parameters of our model. The composite leptoquarks have the same couplings structure as the leptoquarks in the so-called minimal model. The difference in terms on bounds is that in the composite model, R D ( * ) receives approximately equal contributions from the leptoquark and the W H3 boson. Thus in bounding the leptoquark parameter one should rescale the parameter g U of [45] by a factor of 1/ √ 2. In doing so, the bounds on vector leptoquark from τ − τ + searches in the upper panel of figure 6 of [45] are relaxed. 5 Note that we use v ≈ 175 GeV, whereas in [45] one uses v ≈ 250 GeV. |s Lu3 s Lν3 | ≈ 0.7 ÷ 0.8 for ξ = 0.1 and, to be consistent with bb → τ + τ − searches at LHC, relatively large couplings of the composite vectors, g ρ , g ρR 3 ÷ 4, always for ξ = 0.1.

Measurements in
In case the anomalies will persist and perhaps be reinforced in experiments to come, several more detailed investigations can be performed of the model described here to prove its full compatibility with the various constraints, present and future. They include three main chapters: i) electroweak corrections, both of oblique and non-oblique nature, extending and completing sec. IV; ii) flavor physics, as partly already discussed in [2], both in the quark and in the lepton sector; iii) LHC searches, as outlined in sec. V.
Zürich for its hospitality while part of this work was completed. The discussion for the SU (4) extension of the triplet scenario proceeds along the same lines as the bidoublet scenario: the fermionic particle content is where β = 1, 2, 3 is a fundamental color index. Under SU (4) × SU (2) L × SU (2) R × U (1) X they transform like ψ = (4, 2, 2) 1/2 , χ = (4, 1, 3) 1/2 and χ = (4, 3, 1) 1/2 . The components of ψ, χ and χ are A notable difference is that there is only one composite doublet with the same quantum numbers of q L , consequently z 3 = z 12 = 1. This is not a problem for the Zb LbL coupling deviation because with this choice of representations for composite quarks the tree level deviation is zero as can be understood by the symmetry considerations of [58]. Another important difference is that composite states with the same quantum numbers of u R and d R are inside the (1, 3) multiplet of SO(4) ∼ = SU (2) L × SU (2) R and do not live in different multiplets as in the bidoublet case. Therefore, right-compositeness cannot be implemented and we will focus on left compositeness The Yukawa and mass terms for the fermionic resonances in the strong sector are given by L triplet where Y ± , m ψ and m χ ( ) are U (2) preserving flavor diagonal matrices. Quark and lepton mixing Lagrangians are given by The SM Yukawa matrices can be written as in eq.s (24) and (25) with with y b and a d given by the same expressions of y t and a u provided that right up-mixing angles and Y + are replaced by right down-mixing angles and Y − .

Appendix B: Interactions of Composite Vectors with Elementary Fermions
This appendix gives the interaction Lagrangians of composite vectors with elementary quarks and leptons that are relevant for the processes discussed in section III. Each piece of the Lagrangian below contains one composite vector. Explicitly, the individual terms are: Lν3 s 2

Lν3
2 U e * 3i U e 3j ē Li γ µ e Lj +ν Li γ µ ν Lj , Lν3 U e * 3i U e 3j −ē Li γ µ e Lj +ν Li γ µ ν Lj , Lν3 U e * i3 U e 3jν Li γ µ e Lj + h.c.. (B7) The coefficients A (V ) f are defined in eq. (31), and the matrix elements U d,u 3i can be written in terms of CKM matrix elements as The leptoquark Lagrangian, L LQ , makes use of the following definitions where θ l is the angle (s l = sin θ l , c l = cos θ l ) in the unitary transformation which diagonalizes ∆ e on the left side and l ≡ x τ |V|.
In the couplings of V µ and W H± µ we are neglecting terms in the 1−2 sector proportional to the square of the small s L2 mixings. The interactions of W R± are not shown since they always involve at least one exotic fermion.